Solve the following nonlinear equation by an appropriate sub

Solve the following nonlinear equation by an appropriate substitution y\" = 2x(y\')2. Y\" + yy\' = 0, y(0) = 1, y\'(0)= -1

Solution

a)
note that:

y\'=dy/dx and y\"=d²y/dx²

we have
y\" = 2x(y\')²

=> y\"/(y\')² = 2x

integrating both sides:

-1/y\' = x² +c

or say -dx/dy= x²

(-1/x²)dx = dy

again integrating both sides:

1/x = y +c